This talk will discuss: (1) How much time does it take to determine if a curve is knotted? (2) If a curve has length L and is unknotted, what is an upper bound of the area of a disk with boundary the knot? (3) If a polygonal curve has n edges and is unknotted, how many triangles are needed to construct a disk with boundary the knot? These questions are closely connected and have some unexpected solutions. This is a series of joint works with Lagarias, Pippenger, Snoeyink and Thurston.